Control Strategies for Disturbance Rejection in a Solar Furnace

Transcription

1 Milano (Italy) August 28 - September 2, 211 Control Strategies for Disturbance Rejection in a Solar Furnace Manuel Beschi Manuel Berenguel Antonio Visioli Luis José Yebra Dipartimento di Ingegneria dell Informazione, University of Brescia, Italy {manuel.beschi,antonio.visioli}@ing.unibs.it Departamento de Lenguajes y Computación, University of Almería, Spain beren@ual.es CIEMAT-PSA, Tabernas, Almería, Spain luis.yebra@psa.es Abstract: This paper addresses the temperature control problem in a solar furnace. In particular, two control strategies for the rejection of disturbances (represented by the variation of the input energy provided by the Sun, mainly because of passing clouds) are proposed. The firstoneisbasedongeneralisedpredictivecontrol,whereanonlinearmodel isemployedforfree response prediction while a linearised model is used for the computation of the forced response. The second one is a constrained control strategy where the process input and output constraints are taken into account explicitly. In both cases an adaptation of the most significant process parameter is performed. Simulation results show the effectiveness of the methodologies. Keywords: Solar furnace, disturbance rejection, predictive control, constrained control. 1. INTRODUCTION In the last forty yearsthere has been an increasinginterest in solar energy because it represents a green and cheap source of energy. In this context, the material treatment in solar furnaces is one of the most promising applications that have been addressed. A solar furnace allows using solarenergyformaterialtreatconsistsofacollectorsystem with tracking (usually made of flat-faceted heliostat) and a static parabolic system which concentrates a high percentage of the solar energy in its focal spot (Figure 1). In the process considered in this paper (see Section 2) the incoming light (that is, the energy flux entering the furnace) is regulated by means of a computer controlled louvered shutter (Berenguel et al., 1999). This type of testing in a solar furnace usually aims at improving the mechanical properties, such as hardness and wear resistance by means of heating the studied samples under varied temperature patterns. The application considered in this paper is the copper sintering, where a sample is exposed to long high temperature set points (near the melting point), so that small particles bond together and the aggregate shrinks resulting in a decrease of surface area and energy, which allowsforulteriorstudyonanumberoftopics,e.g.,volume diffusion ofatoms (?). Due to the complexity and diversity of sample materials and temperature trends such research Partial support for this work has been provided by an Erasmus Placement scholarship which has funded the visit of M. Beschi to Almería. This work has been developed under project DPI C4-4, funded by Spanish Ministry of Science and Innovation and EU-ERFD.The authors thank personnel of the Plataforma Solar de Almería, especially Lidia Roca and Inmaculada Cañadas. plantsareusuallymanuallycontrolledbyexpertoperators. Obviously, the efficiency of the operations depends on the operators skill and therefore the presence of a properly designed automatic control system would have the advantage of providing adequate results for different operating conditions.forthisreason,differentcontrolstrategieshave been published in literature (see, for example, (Lacasa et al., 26; Berenguel et al., 1999; da Costa and Lemos, 29)). However, the compensation of disturbances can be improved. Indeed, the input energy provided by the sun is variable, because of its daily cycle and because of passing clouds, which represent the most critical issue. This might obviously seriously decrease the performance of an experiment. In this paper two different control strategies for the compensation of disturbances are presented. In both cases the input signal of the process is the aperture of the shutter and the output signal is the temperature measured by a thermocouple welded to the side or the back ofthe sample. The first method is based on Generalised Predictive Control(GPC)(CamachoandBordóns,24)whilethesecond is a constrained control strategy based on a nonlinear control law which takes into account the input and output constraints of the plant. Model predictive control schemes basedonlinearizednonlinearmodelshavebeenextensively used (Qin and Badgwell, 23), but in this paper different approaches are developed. The obtained performance in both cases is compared with that obtained by a standard Proportional-Integral (PI) control. The paper is organised as follows. In Section 2 the solar furnace is briefly described. In Section 3 the GPC-based control system is presented while in Section 4 the con- Copyright by the International Federation of Automatic Control (IFAC) 12243

2 Milano (Italy) August 28 - September 2, 211 trained control strategy is proposed. Simulation results (including the comparison with PI control) are shown in Section 5 while conclusions are drawn in Section DESCRIPTION OF THE SYSTEM The process considered in this paper is the solar furnace (Martínez,1996)ofCIEMAT-PlataformaSolardeAlmería (PSA, which is the largest European center for research, development and testing of concentrating solar technologies located in Tabernas, Almería, South- East Spain. The solar furnace mainly consists of a heliostat which tracks the sun using an azimuth and pitch positioning mechanism and that reflects sunlight onto a a concentrator disk. The amount of incoming energy is modulated using a louvred shutter (control actuator). The concentrator convex mirror gathers most of incoming sunlight from the outdoor heliostat onto a focal spot of 22 cm diameter, located inside a vacuum chamber where samples are placed for thermal tests. Figure 1 shows a representation of the plant layout. dependenceontemperaturedoesnotconsiderablyimprove the obtained results and on-line adaptation of only one parameter is also possible. In this way the parameters can be grouped together and equation (1) becomes: where: [ 1 sin[α ] (1 U/1)] sinα ( G T 4 T 4 Te 4 ) GT (T T e ) T =G U I G T 4 = α eσs s, G T = α cs s ; mc e mc e G U = ρ c ρ h G U, G U = S cs s α a. S f(9%) mc e In practical cases, ρ h and ρ c are measured before a test (typical values are 75% and 95%), while G U, G T 4 and G T can be estimated with a standard least squares method. Resulting values are G U = [m 2 KJ 1 ], G T 4 = [s 1 K 3 ]), and G T = [s 1 ]. The comparison between historical data and the calibrated model shows that the mean absolute percentage error is less than 2%, while the maximum relative error is 7% (Beschi, 21), see Figure 2 as an example. The shutter aperturehasarange[%-1%].inaddition,themaximum velocity of the shutter aperture is limited to 5%/s. In any case, the presence of a vacuum chamber (Minivac) limits the maximum admissible velocity. In fact, a fast shutter aperture can generate a thermal shock in this equipment. The slew rates are therefore set equal to.5%/s for the positive velocity and 2%/s as the negative value. The negative value is greater than the positive one because if there is a large radiation increment it is necessary to close rapidly the shutter to prevent thermal shocks. (2) Fig. 1. PSA Solar furnace layout Thesample stemperaturemodelisdescribedin(berenguel et al.,1999).the modelconsistsofthe followingfirst-order nonlinear equation (1): [ S c ρ h ρ c IS s α a 1 sin[α ] (1 U/1)] S f(9%) sinα ( α e σs s T 4 Te 4 ) αc S s (T T e ) = d(mc (1) et) where: ρ h and ρ c are the reflectivity coefficients of the heliostat and the concentrator [-]; S c, S s and S f(9%) are the surfaces of the concentrator, of the sample and the focus area where the 9% of the solar input energy is concentrated [m 2 ]; I is the input solar radiation [Wm 2 ], which is measured by using a pirheliometer; α a is the absorptioncapacityofthesample[-];α c istheemissivityof thesample[wm 2 K 1 ];α e isthecapacityofthesampleto exchangeheat with the air [-]; σ is the Stephan-Boltzmann constant [ Wm 2 K 4 ]; C e is the specific heat [Jkg 1 K 1 ]; misthe samplemass[kg];α isthe maximum angular aperture of the shutter [rad]; T is the sample temperature [K] and suppose uniform into the body; T e is the environmental temperature [K]; U is the angular percentage aperture [%]. In this work all the physical properties are supposed to be constant, as it has been observed in control tests that taking into account their simulation real data 1: 11: 12: 13: local time (hours) Fig. 2. Comparison: nonlinear model vs. real output 3.1 Basic algorithm 3. GPC-BASED STRATEGY The disturbance rejection task is addressed by means of GPC (Camacho and Bordóns, 24). The control signal is the sum of two terms. The first one is a feedforward actionu th (t)whichisprecalculated,withthehypothesisof perfect modelandasolarradiationequalto thetheoretical valuei th (selectedequalto9wm 2,whichistheaverage annual radiation in Almería), in order to obtain the corresponding output T th (t) equal to the set-point value. If the actual values of the temperature and its derivative are known, the value of the theoretical shutter aperture 12244

3 Milano (Italy) August 28 - September 2, 211 can be obtained in closed form by inversion of the model equation. The second term is the GPC signal u(t), which is calculated on-line using a model linearised around the current operating point (Figure 3). Uth u Plant GPC Model I Ith Fig. 3. Scheme of GPC-based control The GPC strategy utilises a Controller Auto-Regressive IntegratedMovingAverage(CARIMA)model.Thismodel is a good approximation of many single-input singleoutput plants around an equilibrium point: A(z 1 )y(t) = z d B(z 1 ) u(t 1)C(z 1 )e(t) (3) where d is the dead time of the system, y(t) is the output of the plant, u(t) is the control action, e(t) is a zero mean white noise, A, B, C are adequate polynomials in the backward shift operator z 1 and (z 1 ) = 1 z 1 (Camacho and Bordóns, 24). Considering smooth nonlineardynamics(asthoseprovidedby thesolarfurnace model), the output prediction can be approximated by the superposition of the forced and free responses (Camacho and Berenguel, 1994), y = Gu f where y is the vector containing future ahead predictions of the system output on data up to time t, matrix G is built by the process open-loop step response coefficients g i, f represents the plant free response and u is the vector of the future control increments obtained from the solution of the GPC optimization problem described below. To obtain the furnace linearised model, two new variables are introduced: y(t) defined as the error between the plant temperature T(t) and the model temperature T th (t); i(t) defined as the difference between the real radiation I(t) and I th. Using the furnace model shown in Section 2 and the definition of i(t) and y(t) it is possible to write: ( )) sin (α 1 U th(t)u(t) 1 G U (I th i(t)) 1 sinα ( ) G T 4 (T th (t)y(t)) 4 Te 4 (4) G T (T th (t)y(t) T e ) = dt th(t) T Tth dy(t) As in (Berenguel et al., 1999), this equation can be simplified and linearised as: Time Constant (s) Gain ( C/%) Temperature ( C ) Fig. 4. Values of the linearised system parameters for different values of the sample temperature. ( )) sin (α 1 U th(t) 1 G U 1 i(t) sinα α (5) G U I th cos[α (1 U th (t)/1)]u(t) 1sinα ( 4G T 4Tth 3 (t)g ) dy(t) T y(t) = Equation (5) describes the linearised model around the actual operating point. The transfer function between u(t) and y(t) has the following form: Y(s) U(s) = K (6) τs1 where: α K = τg U I th cos[α (1 U th (t)/1)] (7) 1sinα and 1 τ = 4G T 4Tth 3 (t)g. (8) T The values of K and τ resulting for different values of the temperature T th are plotted in Figure 4, where the smooth nonlinearity of the process appears. Notice that for generating the figures the steady-state value of U th for each temperature T th has been used. The transfer function can be discretised as: y(t) = ay(t 1)bu(t 1) with a = e ts/τ and b = K(1 a), where t s is the sampling period. The step response coefficients can be easily determined as: g i = 1 ai1 1 a b. The GPCcostfunction J tobe minimised in this casewith respect to the future control increments is N 2 N u J = [y(tj t)] 2 λ(j)[ u(tj 1 t)] 2 (9) j=n 1 j=1 which represents the sum of two contributions: the first one to tries to minimise the distance between the actual and the theoretical temperature trajectory in a temporal horizon between times N 1 and N 2 ; the second one measures the control effort in a control horizon N u. The two terms are weighted with the parameter λ(j). As the dead time of the process is negligible, suitable values for N 1, N 2 and N u are N 1 = 1 and N 2 = N u = N. The horizon N has to be chosen big enough to consider the complete system dynamics. The parameter λ is usually supposed constant for all the horizon. The sampling period t s has been fixed 12245

4 Milano (Italy) August 28 - September 2, 211 to 3 seconds, which is a sufficiently small value in comparison with the linearised system time constant, which varies betweenabouthalfanhourforlowtemperaturesandafew minutes around 1 o C (see Figure 4). The horizon N has therefore been fixed to 4. The prediction time is smaller than the time constant, but it is necessary to remember that the linearisation is a good approximation only near the operating point and therefore a long term prediction is not sensible. For example, if the solar furnace works with a temperature of 2 o C and it must follow a ramp of 2 o Cmin 1, after the prediction time (two minutes) the time constant decreases by 1%. Moreover, the system is subjected to fast disturbances coming from the solar radiation and thus using a prediction of the solarradiation equal to its actual value would not provide good results if a long prediction horizon is used. The cost function J can be rewritten as (Camacho and Bordóns, 24): ( ) J = u T G T GλI u2f T Guf T f. The constraints considered in the optimisation are the limited amplitude and slew rate of the control signal. In factitisnotpossibletoguaranteetherespectoftheoutput limits because they depend on solar radiation and it is not always possible to compensate its variation because of the actuator s limit. Thus, an unfeasible solution of the optimisation problem may occur. The control signal limits are denoted as U as the minimum limit and U as the maximum limit. They can be written as: l(u u(t 1)) Hu l(u u(t 1)) where H is a lower triangular matrix of ones and l is a column vector of ones. The slew rate limits of the control signal are u and u and their representation is: lu u lu 3.2 Nonlinear prediction of free response In a nonlinear system like the solar furnace the linear approximation is a good model only around the operating point. To obtain a better prediction it would be necessary to use a nonlinear predictive control, but in this way the computational time increases too much. A way to refine the prediction is using a linear model for the forced response, so that the optimisation is still a quadratic programming problem, and to use a nonlinear free response prediction. Although from the theoretical point of view the algorithm is questionable because the superposition theorem is only valid for linear systems, the practical effectiveness forthis approachfor solarplants with smooth nonlinear dynamics has been demonstrated in (Camacho and Berenguel, 1994). To calculate the nonlinear free response prediction, the nonlinear model is simulated considering that along the prediction horizon, the input to the system is U th (t j t) u(t) and the radiation equal to I(t) (last measured value). The theoretical temperature T th (tj t) is subtracted from the resulting temperature to obtain the free response of the error f(tj t), as the GPC workstominimizethedifferencebetweenthemodeloutput and the theoretical output (the output of the model when the input is U th and the radiation is I th ). Notice that U th and T th vary along the horizon. If a temperature profile is fixed, U th is computed by model inversion, while if U th is imposed (for example a ramp until reaching the desired final set-point), the value of T th is computed by using the nonlinear model. For the GPC algorithm U th acts as an exogenous signal (not modifiable by the GPC controller), and in this case the desired trajectory is considered to be known. 3.3 Gain adaptation algorithm All model predictive control strategies have a good performance if the model is sufficiently accurate. If there are model mismatches, a way to improve the performance is to adapt the parameters on-line. This strategy must be applied with care because it is possible that the frequency content of the data is not sufficient for a good estimation. In this work the implemented algorithm is as follows: if the error between the prediction and the actual value is bigger than two Kelvin the parameter G U is set as: G U (t1) = ( 1 T(t) ˆT(t t 1) 1 ) G U (t) where T(t) is the actual plant temperature and ˆT(t t 1) is the last predicted temperature. The parameter can not increase more than the double of the nominal value and it must be greater than the half of the nominal value. This limitation is done for safety reasons. When the prediction error decreases under one Kelvin the parameter keeps its value. It is worth noting that the decision to modify only the parameter G U has been taken because this parameter has the greatest influence on the system performance and therefore it is important to have an accurate estimation of its value. The estimation error of the other parameters does not affect significantly the performance obtained. 4. CONSTRAINED CONTROL This section presents an alternative approach for controlling the thermal process in the solar furnace. The basic idea is to use a standard PI algorithm when the operating point is close to the theoretical one and to use a nonlinear control law when the operating conditions are far from the equilibrium. The nonlinear control law explicitly takes into account constraints on the process output (as well as on the process input). The gain adaptation algorithm proposed for the GPC strategy is used also in this context. 4.1 Switching strategy The switching between the two control algorithms is a critical issue. Indeed, it is necessary to prevent excessive commutations between the algorithms and to guarantee the continuity of the solution. The first problem is solved by using a hysteresis commutation. Thus, the switching from nonlinear to linear PI control occurs when the temperature error is less than 1 o C, viceversa, the switching from linear PI to nonlinear control occurs when the error is greater than 2 o C. The continuity of the control signal is guaranteed by using a integrator after the switch (Lourenco and Lemos, 26). As a consequence the output of both the control strategies must be the first derivative of the desired control action

5 Milano (Italy) August 28 - September 2, PI control The linear control consists of a standard PI controller (which aims at minimising the difference between the theoretical process output and the actual one) with a constant feedforward signal U th, which is devoted to maintain the equilibrium point, that is, the value U th is such as the corresponding output is T th, by assuming a perfect model and a radiation equal to the theoretical value I th (Figure 5). A second feedforward action is devoted to the disturbance compensation. Indeed, the continuous system transfer function (6)-(7) linearly depends on the solar radiation. Thus, as shown in (Berenguel et al., 1999), it is convenient to place the feedforward block I th /I(t) in the forward path (i.e., at the output of the PI controller) so that the resulting global transfer function is theoretically independent of the solar radiation values. Uth u Plant Linear Control Model I Ith T Tth Fig. 5. Scheme of PI control for the solar furnace By denoting the difference between the theoretical output and the actual one as y(t), the control law expression is: u(t) = I t ] th [K p (y(t)1/t i y(v)dv )U th I(t) t where K p is the proportional gain and T i is the integral time constant. The first derivative of the control action, which is necessary for the switching algorithm, is: u(t) = I ( th [K p ẏ(t) 1 )] y(t) I thi(t) I(t) T i I(t) 2 u(t) The tuning of the PI parameters have been performed by means of an iterative numerical algorithm that minimises the averagevalue ofintegrated absolute errorsobtained by consideringtheloaddisturbanceresponseinthreedifferent cases, that is, at three different set-point temperatures. 4.3 Nonlinear control In order to increase the performance also when the system is far from the operating point, a nonlinear control law has been devised. The rationale of this strategy is to select the desired derivative functions of the output variable and to obtain the control variable by the inversion of the inputoutput equation. Consider a first-order nonlinear system described by the equation: ẏ(t) = f(y(t),u(t),d(t)) (1) where d(t) is vector of the nonmanipulated inputs. If f(y,u,d) is invertible, it is possible to write: u(t) = g(y(t)),ẏ(t),d(t)). By derivating with respect to the time it is possible to obtain the signal u(t) which is necessary for the switching algorithm: u(t)= g g (y(t),ẏ(t),d(t))ẏ(t) y ẏ (y(t),ẏ(t),d(t))ÿ(t) g (y(t),ẏ(t),d(t)) ḋ(t) (11) d The choice of ẏ(t) and ÿ(t) can be done by plant constraints, empirical considerations or operators experience. The instantaneous values of ẏ(t) and ÿ(t) are chosen with the following procedure, where ẏ M and ÿ M are the maximum limits of the derivatives: (1) The last output derivative is calculated by the equation (1) performed at the time t t s ; (2) if y(t) y th (t) < (a) if ẏ 2 (t t s ) 2ÿ M y(t) y th (t) then ÿ(t) is set equal to ÿ M ; (b) elseif ẏ(t t s ) < ẏ M then ÿ(t) is set equal to ÿ M ; (c) elseif ẏ(t t s ) > ẏ M then ÿ(t) = ÿ M ; (d) else ÿ(t) is set to zero; else (a) if ẏ(t t s ) 2 2ÿ M y(t) y th (t) then ÿ(t) is set equal to ÿ M ; (b) elseif ẏ(t t s ) > ẏ M then ÿ(t) is set equal to ÿ M ; (c) elseif ẏ(t t s ) < ẏ M then ÿ(t) is set equal to ÿ M ; (d) else ÿ(t) is set to zero; (3) the desidered first derivative is set as ẏ(t) = ẏ(t t s )ÿ(t)t s (4) the derivative of the control action is calculated by expression (11). (5) if u(t) exceeds the limits, then it must be saturated. In the solar furnace of Almería, the operators suggest 4 o C/min as maximum peak of temperature s derivative, the second derivative peak is chosen to 12 o C/min SIMULATION RESULTS Theproposedcontrolmethodologieshavebeen tested with a set-point temperature of 2 o C, 6 o C and 1 o C, in order to highlight the performance achieved in different operating points. In these cases the values of the linearised system time constant are respectively (see (8)) 178 s, s, and s. The simulations have been made by using the radiation of a cloudy day and, in addition, constant and variable errors have been introduced in the values of the model parameters. In particular, the mean errorsofthe parametersg U, G T and G T 4 arerespectively: 5%, 1% and -2%. The variable errorshave a normal distribution with standard deviation of 1%. The two methodologies proposed in the previous sections are compared with a standard PI controller with the parameters tuned according to the optimisation technique described in subsection 4.2. Figures (6)-(8) represent the evolution of temperature, shutter aperture and radiation during the simulations, for the three considered cases. It appears that the GPC provides the fastest settling time, while the constrained control strategy has, as expected, a greater limitation of temperature derivative, because of the selected output constraints. This aspect is considered 12247

6 Milano (Italy) August 28 - September 2, Shutter aperture (%) Shutter aperture (%) Radiation (W/m²) Fig. 6. Simulation results for the set-point temperature of 2 o C. Solid line: PI controller. Dashed line: GPC. Dash-dot line: constrained control. Radiation (W/m²) Fig. 8. Simulation results for the set-point temperature of 1 o C. Solid line: PI controller. Dashed line: GPC. Dash-dot line: constrained control. 3 Shutter aperture (%) Radiation (W/m²) Fig. 7. Simulation results for the set-point temperature of 6 o C. Solid line: PI controller. Dashed line: GPC. Dash-dot line: constrained control. important by the furnace operators. In fact, in this way the equipment presents a smaller thermal stress. Figure 9 represents the adaptation of the gain G U with the constrained control strategy for the case of the setpoint temperature of 1 o C. Similar results are obtained in the other cases and for the GPC-based method. It can be noted that the value of the parameter is conveniently updated in order to compensate for the estimation error. 6. CONCLUSIONS In this paper two control methodologies have been proposed for the disturbance rejection in a solar furnace. While the GPC-based control provides the best settling time in general, the constrained control method presents the remarkable feature that the constraints on the process input and output are satisfied, thus avoiding a possibly dangerous thermal stress of the material. Future work will include the output constraints in the GPC controller (treatment of unfeasibility problems caused by solar radiation variations through reference governor approaches) G U increment (%) 2 1 Fig. 9. Adaptation of the gain in the simulation with the constrained control strategy and with a set-point temperature of 1 o C. as far as experiments on the real solar furnace in order to verify the effectiveness of the control techniques in practical cases. REFERENCES Berenguel, M., Camacho, E., García-Martín, F., and Rubio, F. (1999). Temperature control of a solar furnace. IEEE Control System Magazine, 19(1), Beschi, M. (21). Constrained control of the temperature of a solar furnace. Master Thesis, University of Brescia. Camacho, E. and Berenguel, M. (1994). Application of generalized predictive control to a solar power plant. In IEEE International Conference on Control Applications. Glasgow, UK. Camacho, E. and Bordóns, C. (24). Model Predictive Control. Springer, London, UK. da Costa, B.A. and Lemos, J. (29). An adaptive temperature control law for a solar furnace. Control Engineering Practice, 17, Lacasa, D., Berenguel, M., Yebra, L., and Martínez, D. (26). Coppersinteringinasolarfurnacethroughfuzzy control. In IEEE International Conference on Control Applications. Munich, Germany. Lourenco, J. and Lemos, J. (26). Learning in switching multiple model adaptive control. IEEE Instrumentation & Measurement Magazine, Martínez, D. (1996). Solar Furnace Technologies. CIEMAT. Qin, S. and Badgwell, T. (23). A survey of industrial model predictive control technology. Control Engineering Practice, 11,